Диссертация (Isomonodromic deformations and quantum field theory), страница 10

We define the vertex operators in this theory in terms of solutions of thecorresponding isomonodromy problem. We use this construction to get some newinsights on tau-functions of the multicomponent Toda type hierarchies for the class ofsolutions, given by the isomonodromy vertex operators and get useful representationfor the tau-function of isomonodromic deformations.IntroductionThe aim of the chapter is to present briefly the main free-fermionic constructions thatappear in the study of correspondence between the problem of isomonodromic deformations and two-dimensional conformal field theories – for some class of the theorieswith extended conformal symmetry.

An interest to the two-dimensional conformalfield theories (CFT) with extended nonlinear symmetries, generated by the higherspin holomorphic currents, has been initiated by pioneering work [ZamW]. Thesetheories with so called W-symmetry possess many features of ordinary CFT, including the free field representation [FZ, FL], which becomes especially simple for thecase of integer Virasoro central charges. However, even in this relatively simple caseit turns already to be impossible to construct in generic situation the W-conformalblocks [BW], which are the main ingredients of the conformal bootstrap definition ofthe physical correlation functions [BPZ].This interest has been seriously supported already in our century by rather nontrivial correspondence between two-dimensional CFT and four-dimensional supersymmetric gauge theory [LMN, NO, AGT], where the conformal blocks have to becompared with the Nekrasov instanton partition functions [Nek, NP] producing inthe quasiclassical limit the Seiberg-Witten prepotentials [SW].

This correspondencealso meets serious problems beyond SU (2)/Virasoro level: both on four-dimensionalgauge theory and two-dimensional CFT sides. These difficulties can be attackedusing different approaches, for example in [GMtw] we have demonstrated how the373. Free fermions, W-algebras and isomonodromic deformationsexact conformal blocks for the twist fields [ZamAT87, ZamAT86, ApiZam] in theories with W-symmetry can be computed, using the technique developed previously in[KriW, Mtau, GMqui].Here we present another approach to the study of the CFT vertex operators in thetheories with extended conformal symmetry, based on their free-fermionic construction.

It is clear, that it should work (at least) in the cases of integral central charges,where it is intimately related with the recently discovered there CFT/isomonodromycorrespondence [GIL12, Gav]. We are going to discuss the operator content of thesetheories with nontrivial monodromy properties, and then turn to the problem of computation of the matrix elements of generic monodromy operators. Finally, we aregoing to relate these matrix elements with the tau-functions of two different classesof problems – the tau-functions of the multicomponent classical integrable hierarchiesof Toda type, and the tau-functions of the isomonodromic deformations.Abelian U (1) theoryFermions and vertex operatorsIntroducestandard two-dimensional holomorphic fermionic fields with the action´ the12S = π Σ d z ψ̃∂ψ, so thatψ̃(z)ψ(z 0 ) =1+ ...z − z0(3.1)or{ψr , ψ̃s } = δr+s,0 ,X ψr,ψ(z) =z r+1/21r, s ∈ Z + 21 ,Xψ̃(z) =s∈Z+ 12r∈Z+ 2ψ̃s(3.2)z s+1/2with the half-integer mode expansion.

The bosonization formulas readψ̃(z) =: eiφ(z)−:= ePn<0Pψ(z) =: e−iφ(z):= en<0Jn −nznJn −nznP−en>0Pen>0Jn −nznJn −nznee Q z J0 ,−Q −J0z(3.3),whereJ(z) =: ψ̃(z)ψ(z) := i∂φ(z) =[Jn , Jm ] = nδn+m,0 ,n, m ∈ Z,X Jn,z n+1n∈Z(3.4)[Jn , Q] = δn0 ,where normal ordering means, that all negative modes stand to the left of all positive,and all Q to the left of J0 .Consider now generic vertex operators for the bosonic fieldsiνφ(z)Vν (z) =: e−ν:= ePn<0Jn −nzn−νePn>0Jn −nzn38eνQ z νJ0 ≡ Vν− (z)Vν+ (z)eνQ z νJ0(3.5)3.2.

Abelian U (1) theorywhich satisfy the obvious exchange relations, following from the Campbell-Hausdorffformulaw αβ −Vβ (w)Vα+ (z) ,Vα+ (z)Vβ− (w) = 1 −z(3.6) z αβ w αβ z −αβVα (z)Vβ (w) =1−1−Vβ (w)Vα (z) .wzwOne can also writeVα (z)Vβ (w) = (z − w)αβ : Vα (z)Vβ (w) : .(3.7)Since vertex operators contain the factor eνQ , they shift the vacuum chargeVν (z) : Hσ → Hσ+νwhen acting onto a sector in full Hilbert spaceMH=Hσ(3.8)(3.9)σcorresponding to the definite value of this charge. Notice that we do not impose anyspecial constraints to the (real) values of the vacuum charges σ ∈ R.The Hilbert space Hσ is constructed by the action of the negative bosonic generatorsJ−n1 .

. . J−nk |σi(3.10)on the vacuum vector J0 |σi = σ|σi, and these states can be labeled by the Youngdiagrams with the row lengths n1 , . . . , nk .One can also construct the action of the fermionic operators on this vector space.Then the bosonization formulas (3.3) will generally produce the fractional powers inholomorphic coordinate z due to the factors z J0 , while e±Q just shift the vacuumcharge by ±1.

It means that one can define the (multiple) action of the modes of theoperatorsX ψ̃ σX ψσrrσ,ψ̃(z)=ψ σ (z) =(3.11)r+1/2+σr+1/2−σzzrrin the direct sum of the Hilbert spacesHσ =MHnσ(3.12)n∈Znaturally labeled by some fractional σ ∈ R/Z.Basis in the each space Hnσ can be given by the vectors generated by the zero-chargeexpressions of the fermionic modes. As in bosonic representation, these vectors canbe labeled by the Young diagramsYσ|Y, σi =ψ̃−pψ σ |σii −qi(3.13)iwhere now pi and qi are the Frobenius coordinates of the Young diagram. In ourconvention they are half-integer, and can be easily read of the following picture:393.

Free fermions, W-algebras and isomonodromic deformations@@@i.e. one has to cut the diagram by the main diagonal and just take the areas of therows and columns starting from the diagonal cells. For example, the Young diagramfrom the picture has {pi } = { 29 , 52 , 23 } and {qi } = { 92 , 52 , 21 }.The states in the dual to Hσ module can be obtained by the Hermitian conjugationYhσ, Y | = hσ|ψ̃qσi ψpσi .(3.14)iOur main aim in what follows is to compute the matrix elements of the operatorVν (1) = Vν between the arbitrary fermionic statesZ(ν|Y 0 , Y ) = hθ + ν, Y 0 |Vν (1)|Y, θi .(3.15)The most straightforward way is to use explicit bosonic representation (3.5) of thevertex operatorYYψ̃−pi ψ−qi |σi =ψ̃qj0 ψp0j Vν− Vν+ eνQZ(ν|Y 0 , Y ) = hσ + ν|ij= h0|Y−V−νψ̃qj0 Vν−·−V−νψp0j Vν−+Vν+ ψ̃−pi V−ν+· Vν+ ψ−qi V−ν|0i =(3.16)ij= h0|YYY(Vν− )−1 ψ̃qj0 Vν− · (Vν− )−1 ψp0j Vν−Vν+ ψ̃−pi (Vν+ )−1 · Vν+ ψ−qi (Vν+ )−1 |0i .jiIt is easy to understand from (3.3) and (3.5) that the consequent triple products ofoperators in this formula can be considered as certain adjoint action, or just conjugations of the fermions, which turn under such action just into the linear combinationsof themselves.

At the level of generating functions it looks likeVν+ ψ̃(z)(Vν+ )−1 = (1 − z)ν ψ̃(z) ,ν1− −1−ψ̃(z) ,(Vν ) ψ̃(z)Vν = 1 −zVν+ ψ(z)(Vν+ )−1 = (1 − z)−ν ψ(z) ,−ν(3.17)1− −1−(Vν ) ψ(z)Vν = 1 −ψ(z) ,zor, more generallyVν (w)−1 ψ̃ σ+ν (z)Vν (w) =−1Vν (w) ψσ+ν(z)Vν (w) = z νw z −νwX0 1 z nexp νn wnn∈Z!X0 1 z nexp −νn wnn∈Zψ̃ σ (z) ,(3.18)!σψ (z) ,where the formal series in the r.h.s.

can be rewritten with the help of the Fouriertransformation as!X0 z nsin πν X z kexp ν=.(3.19)nπk+νn∈Zk∈Z403.2. Abelian U (1) theoryThis is a particular case of transformations from GL(∞), realized byXars : ψ̃−r ψs :∈ gl(∞),ars → ∞, |r − s| → ∞ ,(3.20)moreover, corresponding to the situation, when ars = Par−s (a well known example ofsuch transformation is generated by the currents Jn = r : ψ̃r ψn−r : from (3.4)). It istrue in the most general case: if one computes any matrix elements of such operator,they always can be expressed in terms of those with only two extra fermion insertions,i.e. we do not need an explicit form of the operator Vν = Vν− Vν+ – just the only factof the adjoint action, and we are going to use this property in more complicated nonAbelian situation below.In particular, one can compute (3.16) first using the Wick theorem!0 ψp0 Vν |σi0 Vν ψ−q |σihσ+ν|ψ̃hσ+ν|ψ̃qqijjjZ(ν|Y 0 , Y ) = det= det Gν (3.21)−hσ + ν|ψp0j Vν ψ̃−pi |σi hσ + ν|Vν ψ̃−pi ψ−qi |σiand then to apply (3.17) to the matrix elements in (3.21).Matrix elements and Nekrasov functionsThe two-fermion matrix elements of the matrix G = Gν (its rows are labeled by{xa } = {qj0 } ∪ {−pi }, whereas columns are labeled by {yb } = {p0j } ∪ {−qi }, here wedenote by p and q some positive half-integer numbers) are expressed asq 0 − 21G(q 0 , p0 ) = h0|ψq0 ψ̃p0 Vν− |0i =X (ν)m (−ν)p0 +q0 −m,0 + q 0 − m)!m!(pm=01q− 2X(−ν)n (ν)p+q−n+G(−p, −q) = h0|Vν ψ−p ψ̃−q |0i =,n!(p+q−n)!n=0p0 − 21G(−p, p0 ) = −h0|ψ̃p0 Vν− Vν+ ψ−p |0i = −(3.22)X (−ν)m (ν)m+p−p0,0 )!m!(m+p−pm=01q− 2X(−ν)n (ν)n+q0 −q0− +G(q , −q) = h0|ψq0 Vν Vν ψ̃−q |0i =.0 − q)!n!(n+qn=0These expressions are easily computed, using adjoint action (3.17) for the componentsVν+ ψ−p (Vν+ )−1∞X(ν)m=ψ−p+m ,m!m=0(Vν− )−1 ψq Vν−∞X(ν)m=ψq−m ,m!m=0Vν+ ψ̃−q (Vν+ )−1(Vν− )−1 ψ̃p Vν−41∞X(−ν)m=ψ̃−q+mm!m=0∞X(−ν)m=ψ̃p−mm!m=0(3.23)3.

Free fermions, W-algebras and isomonodromic deformationswith (ν)m = ν(ν + 1) . . . (ν + m − 1), (ν)0 = 1, and there are explicit formulas for thesums in the r.h.s. of (3.22)bX(ν)b+1 (−ν)a−b(ν)m (−ν)a−m=m! (a − m)!νab!(a − b − 1)!m=0(3.24)bX(−ν)m (ν)a+m(−ν)b+1 (ν)a+b+1=−m! (a + m)!ν(a + ν)b!(a + b)!m=0which can be easily proven by induction. It allows to rewrite matrix elements (3.22)in the factorized form(ν)q0 + 1 (−ν)p0 + 1122,ν(p0 + q 0 ) (q 0 − 21 )!(p0 − 12 )!(ν)p+ 1 (−ν)q+ 1122G(−p, −q) = −,ν(p + q) (p − 21 )!(q − 12 )!(ν)p+ 1 (−ν)p0 + 1122G(−p, p0 ) =,ν(p − p0 + ν) (p − 12 )!(q 0 − 21 )!(ν)q0 + 1 (−ν)q+ 1122.G(q 0 , −q) = − 0110ν(q − q + ν) (q − 2 )!(q − 2 )!G(q 0 , p0 ) =(3.25)The determinant from (3.21) can be therefore written asdet G(xa , yb ) =a,bY (−ν)p0j + 1 (ν)qj0 + 1 Y (ν)pi + 1 (−ν)qi + 1222jν(p0j − 21 )!(qj0 − 12 )!i2ν(pi − 21 )!(qi − 12 )!· det G̃(x̃a , ỹb )a,b(3.26)where now for two new sets {x̃a } = {qj0 } ∪ {−pi − ν}, {ỹb } = {−p0j } ∪ {qi − ν}G̃(x̃a , ỹb ) =sgn(x̃a ỹb ),x̃a − ỹb(3.27)and the corresponding determinant can be computed using the Cauchy determinantformulaQQ1a<b (x̃a − x̃b )a>b (ỹa − ỹb )Qdet=,a,b x̃a − ỹbab (x̃a − ỹb )so one gets finallyZ(ν|Y 0 , Y ) = ±Y (−ν)p0j + 1 (ν)qj0 + 1 Y (ν)pi + 1 (−ν)qi + 122221111 ×00ν(p−)!(q−)!ν(p−)!(q−)!iijj2222jiQQQQQ 0Q 00000i>j (pi − pj )i<j (pi − pj )i>j (qi − qj )i<j (qi − qj )ij (qi + pj + ν)ij (pi + qj − ν)Q 0QQ 0Q×00ij (pi + qj )ij (pi + qj )ij (qi − qj + ν)ij (pi − pj + ν)(3.28)It is easy to see that this expression has the structureZ(ν|Y 0 , Y ) = ±Zb (ν|Y 0 , Y )11Z02 (Y 0 )Z02 (Y )42(3.29)3.2.

Abelian U (1) theorywhere12Z0 (Y ) =YQpi − ! qi − 12 ! Q12iij (pi+ qj )Q,i<j (qi − qj )i<j (pi − pj )(3.30)whileZb (ν|Y 0 , Y ) =Yν −1 (−ν)p0i + 1 (ν)qi0 + 12Y2iν −1 (−ν)qj + 1 (ν)pj + 1 ×22jQ 0Q 0(pi + qj − ν)ij (qi + pj + ν)Q ij 0×Q 0.ij (qi − qj + ν)ij (pi − pj − ν)In this normalization one can check thatYYZb (ν|Y 0 , Y ) = ± (1 + aY (t) + lY 0 (t) + ν)(1 + aY 0 (s) + lY (s) − ν)(3.31)(3.32)s∈Y 0t∈Yis exactly the Nekrasov bi-fundamental function of the U (1) gauge theory at c = 1 or1 + 2 = 0. Notice also thatY11(1 + aY (s) + lY (s))2 = ZV (Y )−1Zb (0|Y, Y ) = Z02 (Y )Z02 (Y ) =(3.33)s∈Yis Nekrasov function for the pure U (1) gauge theory, which corresponds to the Plancherelmeasure on partitions [LMN].Riemann-Hilbert problemThe following simple observation is extremely important for our generalizations below.Consider the correlatorhθ|Vν (1)ψ̃ σ (z)ψ σ (w)|σi = δθ,σ+νz σ w−σ (1 − z)ν (1 − w)−νz−w(3.34)which is easily computed using bosonization rules (3.3).